L(s) = 1 | + (0.00402 + 1.41i)2-s + (−1.09 + 1.09i)3-s + (−1.99 + 0.0113i)4-s + (0.721 + 2.11i)5-s + (−1.55 − 1.54i)6-s + (−2.63 − 0.264i)7-s + (−0.0241 − 2.82i)8-s + 0.595i·9-s + (−2.99 + 1.02i)10-s − 2.35i·11-s + (2.18 − 2.20i)12-s + (−1.27 + 1.27i)13-s + (0.363 − 3.72i)14-s + (−3.11 − 1.53i)15-s + (3.99 − 0.0455i)16-s + (−1.60 + 1.60i)17-s + ⋯ |
L(s) = 1 | + (0.00284 + 0.999i)2-s + (−0.633 + 0.633i)3-s + (−0.999 + 0.00569i)4-s + (0.322 + 0.946i)5-s + (−0.634 − 0.631i)6-s + (−0.994 − 0.100i)7-s + (−0.00853 − 0.999i)8-s + 0.198i·9-s + (−0.945 + 0.325i)10-s − 0.711i·11-s + (0.629 − 0.636i)12-s + (−0.354 + 0.354i)13-s + (0.0972 − 0.995i)14-s + (−0.803 − 0.395i)15-s + (0.999 − 0.0113i)16-s + (−0.389 + 0.389i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.703 + 0.710i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.208364 - 0.499557i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.208364 - 0.499557i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.00402 - 1.41i)T \) |
| 5 | \( 1 + (-0.721 - 2.11i)T \) |
| 7 | \( 1 + (2.63 + 0.264i)T \) |
good | 3 | \( 1 + (1.09 - 1.09i)T - 3iT^{2} \) |
| 11 | \( 1 + 2.35iT - 11T^{2} \) |
| 13 | \( 1 + (1.27 - 1.27i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.60 - 1.60i)T - 17iT^{2} \) |
| 19 | \( 1 + 2.48iT - 19T^{2} \) |
| 23 | \( 1 + (-0.615 + 0.615i)T - 23iT^{2} \) |
| 29 | \( 1 - 0.914T + 29T^{2} \) |
| 31 | \( 1 - 5.53iT - 31T^{2} \) |
| 37 | \( 1 + (7.02 - 7.02i)T - 37iT^{2} \) |
| 41 | \( 1 + 5.48iT - 41T^{2} \) |
| 43 | \( 1 + (-7.03 - 7.03i)T + 43iT^{2} \) |
| 47 | \( 1 + (8.82 - 8.82i)T - 47iT^{2} \) |
| 53 | \( 1 + (-4.92 - 4.92i)T + 53iT^{2} \) |
| 59 | \( 1 - 5.85iT - 59T^{2} \) |
| 61 | \( 1 - 1.78T + 61T^{2} \) |
| 67 | \( 1 + (-5.48 + 5.48i)T - 67iT^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 + (-2.44 - 2.44i)T + 73iT^{2} \) |
| 79 | \( 1 - 8.51iT - 79T^{2} \) |
| 83 | \( 1 + (6.43 - 6.43i)T - 83iT^{2} \) |
| 89 | \( 1 + 8.47T + 89T^{2} \) |
| 97 | \( 1 + (-13.8 + 13.8i)T - 97iT^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.62009724389054899323262666632, −11.24183931239000677538520076204, −10.39306561637895660280825028885, −9.705487576317150077309319567455, −8.653076637147217797924802520082, −7.26223195573428194927845721507, −6.45461667627746173878093832491, −5.67408325498249460911143763479, −4.46276204212709611909727683571, −3.13467626416293569004473283445,
0.43433089791539338018898612534, 2.06008735719514852250263742968, 3.74466204766978232512923171023, 5.10107234797730427022548028586, 6.03002008106654843623737858120, 7.32730328408895046251362289123, 8.733064742599799737515228849818, 9.565260773281333090269029177168, 10.23716962511974300754634979143, 11.62913620338064462156779826471